Double Lie algebroids and representations up to homotopy

Abstract

We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.

Vector bundles over Lie groupoids and algebroids

“…It would be interesting to use this theorem to extend the discussion in Remark 3.1.5 to the context of representations up to homotopy encoded by double Lie algebroids, as studied in [12].…”

On LA-Courant Algebroids and Poisson Lie 2-Algebroids

“…We explain again along the way the parallels between the theory of Lie algebroids, double Lie algebroids and 2-representations on the one hand, and Courant algebroids, LA-Courant algebroids and Lie 2-algebroids on the other hand. We find in particular that a matched pair of 2-representations [5] not only defines a split Lie 2-algebroid [12], but also a split Poisson Lie 2-algebroid-this is in general a different construction. Note here that the five equations that we find show that in the chosen splitting of the underlying Lie 2-algebroid, the Poisson structure defines a morphism from the coadjoint to the adjoint representation up to homotopy of the Lie 2-algebroid [13].…”

Dirac groupoids and Dirac bialgebroids

“…Remark 5.4. A long but straightforward computation shows that these two 2term representations up to homotopy form a matched pair [15]. One of the seven equations defining the matched pair is exactly (1.18) in Lemma A.4, which is also the key to the most complicated equation.…”

“…As a consequence, (D A , U, A, M ) is a double Lie algebroid [15], and the core K has an induced Lie algebroid structure, which is given by the restriction of [· , ·] d to Γ(K), with Lie algebroid morphisms to A and U . To see this, use [15, Remark 3.5] and Lemmas A.1 and A.3 below.…”